Flows and Decompositions of Games: Harmonic and Potential Games

Abstract

In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games.Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to "nearby" games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games.

Four classes of interactions for evolutionary games

“…In opposition to the cross-dependent payoffs we can distinguish games with self-dependent payoffs when the payoff matrices are composed of uniform rows. The corresponding elementary games are defined as g(5) = e (2) ⊗ e (1) , (21) g(6) = e (3) ⊗ e (1) , (22) g(7) = e (4) ⊗ e (1) , (23) and the subset of games with self-dependent payoff can be given as…”

“…5 shows another universal critical transition occurring when the pair interaction is defined by the matrix A = f (12) + f (23) + f (13) . In that case the corresponding potential matrix has three equivalent maximal values (V 11 = V 22 = V 33 ) that prescribes the existence of three (equivalent) homogeneous ordered states in the limit Monte Carlo data for the strategy frequencies versus noise on the square lattice when the pair interaction is defined as A = f (12) + f (23) + f (13) . Symbols agree with those used in Fig.…”

“…[23,24] without the introduction of a concrete set of basis games. The introduction of a suitable set of the orthogonal basis games, however, gives us a more sophisticated knowledge on the anatomy of matrix games.…”

Evolutionary games combining two or three pair coordinations on a square lattice

“…Strategy profiles with maximal U (s) correspond to preferred Nash-equilibria (ground states in the terminology of statistical physics). All matrices that define potential games are linear compositions of cross-dependent, self-dependent, and coordinationtype games [12,[15][16][17]. These game classes represent nonstrategic environmental effects, self-determination, and player-player interactions, respectively.…”

“…The concept of matrix decomposition [12,[15][16][17][18] has revealed that payoff matrices, which define player incomes in two-player games [11,12,19,20], are linear combinations of basis matrices that form four orthogonal subsets and represent different types of interactions: self-dependence, cross-dependence, coordination, and cyclic dominance. These game classes are clearly distinguishable via a few key matrix properties.…”